The Struggle Is Real: Productive Struggle

Learning By Doing

Let's get this party started with a fun little puzzle. You may have seen this on your favorite social media platform [1]. 

See if you can solve it.

Struggling Productively

That was a difficult problem, right? Would you say you "struggled" while attempting to solve it? I know I did! This experience raises a couple of questions:

1. What, exactly, causes us to struggle?

2. When does struggling assist learning, and when does it harm learning? 

3. Under what conditions does struggling lead to long-term learning and transfer? 

Before we attempt to answer these questions, let's acknowledge the emotional components of struggling. I can only speak for myself, but phenomenologically, it doesn't always feel great. I get hot. I feel dumb. Intrusive thoughts distract me from the task-at-hand. Of course, if your working memory is loaded with these intrusive thoughts, then you have fewer resources to dedicate to the current task...which will ultimately cause you to fail!

Sources of Struggle

What causes us to struggle? 

Prior Knowledge: First, we may not have all of the prerequisite knowledge to solve the problem. Once we recognize this fact (hopefully earlier rather than later!), then we can treat it as a quest to find the missing information. 

Incorrect Assumptions: Another reason why we might struggle is because we've made an incorrect assumption. If you assume that a fox and wolf weigh the same, then you will eventually run into a road-block when solving the above problem. 

Unnecessary Problem Constraints: We might struggle because we imposed an unnecessary constraint on the problem. This often happens while solving an insight problem. For example, in the classic nine-dot problem, problem solvers unnecessarily add the constraint that they are not allowed to go beyond the edge of the box. 

Flawed Representation: Finally, we may have chosen a flawed or limited representation. We've seen time and again how important representations are for solving problems. For example, students fail to calculate the area of a parallelogram when presented outside of the canonical orientation (i.e., laying flat along the long side).

What makes struggling "productive?"

According to James Hiebert and Douglas Grouws's book chapter [2], productive struggle happens when a student is working on a problem just outside of their current ability level. This is related to Lev Vygotsky's idea of the "zone of proximal development" (see Fig. 1). 

There are things that you can do autonomously. These are well-practiced skills or declarative knowledge that you've mastered previously. There is also a bunch of things you can't do (at least not yet!). But in between those two spheres are things that you can do with some assistance.

Struggle is most productive when done under the watchful eye of a more knowledgeable partner. They are there to step in and nudge the novice in the right direction. 

Figure 1. Vygotsky's three zones, with the middle as the "zone of proximal development."

Struggle, then, is maximally helpful for several reasons. 

It allows students to appreciate the context of the lesson. If you just give a lecture on solving systems of equations, then the student may not have any appreciation for why systems of equations is a powerful problem-solving technique. However, if you first let them try to figure out how to calculate the weight of the chicken, fox, and wolf, then they might see the utility of systems of equations.

If a student is lacking a key piece of information, then struggling to solve a problem may expose a gap in their knowledge. An impasse in problem solving might force a student to confront the possibility that there is something wrong with their understanding [3]. 

There may be some small amount of discovery involved. For example, there are are (at least) three key insights when solving this animal weight problem: using variables (x, y, & z) instead of animals, isolating a variable for each of the three known weights, and finally substituting the isolated variables into the other equations. Having a key insight or making a discovery is highly motivating.

That brings us to the final reason. Struggle is useful because it necessarily engages a student's conceptual understanding. For any of the three key insights, there is a rich conversation that can connect back to knowledge that a student already possesses (e.g., the concept of a "variable," isolating a variable, variable substitution, and mathematical equivalence). 

The Classroom Connection

Struggling doesn't have to be fraught with negative emotions. In some contexts, struggling is actually kind of fun. Think about the last video game you played. Games are specifically designed to cause you to struggle. In fact, there is some research to suggest that players actually enjoy dying (the ultimate failure!) in first-person shooter games more than shooting other players [4]. Another example is a well-written mystery. You may struggle to figure out "who dun it," but it is an entirely enjoyable experience. It would be beneficial to everyone if academic tasks that cause us to struggle to be structured in a way that is more like a game, puzzle, or mystery. 

Perhaps our struggle as educators and instructional designers, is to figure out how to make struggling an enjoyable educational experience! 🧩


Share and Enjoy!

Dr. Bob

Going Beyond the Information Given

[1] I adapted this problem from Sara Van Der Werf's blog, and you can follow her on twitter @saravdwerf. This might also be a good time to link back to our prior conversation about problem isomorphs.

[2] Hiebert, J., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students’ learning. Second handbook of research on mathematics teaching and learning, 371-404.

[3] VanLehn, K. (1988). Toward a theory of impasse-driven learning. In Learning issues for intelligent tutoring systems (pp. 19-41). Springer, New York, NY.

[4] Ravaja, N., Turpeinen, M., Saari, T., Puttonen, S., & Keltikangas-Järvinen, L. (2008). The psychophysiology of James Bond: Phasic emotional responses to violent video game events. Emotion, 8(1), 114-120. https://doi.org/10.1037/1528-3542.8.1.114

The Mental Tool Rack: Scripts and Schemas

I was in the middle of writing the next great American novel when my hard-drive became extremely fragmented, and it mixed up all of my sentences. Can you help me put them back in order?

1. I cleaned up my mess and threw away my napkins, wrappers, and cup.
2. I waited for my food to be prepared.
3. I ate my delicious food.
4. I paid for my food.
5. I read the menu behind the cashier and decided what I wanted to order.
6. I sat down.
7. I placed my order.
8. I opened the door and walked into the restaurant. 
9. I left the restaurant.

How did you do? When you first started the task, was there any ambiguity that you had to resolve before putting the sentences in order? How did you decide where to put things in sequence? To see the order that I intended, check the footnotes [1].

"Line!" --Ned Nederlander in Three Amigos!

One of the emerging themes throughout this blog is that the mind craves (at least) two things: meaning and order. To make sense of the information-rich world, it helps when actions and events follow a pattern. Over many exposures to the same sequence of events, the mind assembles a script, that is, a mental representation of the expected sequence of events for that situation. You have many different scripts, like going to a restaurant, flying to another city, going to work/school, shopping for groceries, or attending a sporting event. Scripts help set expectations about what is supposed to happen, and they greatly decrease the amount of problem solving that you have to do. If you don't have a script for a given situation, then you find yourself trying to figure out what to do next. 

For example, when my wife and I bought our first car together, we went through a dealership. Neither of us had ever bought a car from a dealer, so we were pretty clueless about all the steps that stood between us and actually driving off the lot in our new (to us) car. We were particularly bewildered when, after we had test driven a couple cars and selected the one we wanted, the salesperson who had been helping us through the process had to go talk to his manager several times during the negotiation process. We now know that this was all part of a very typical script that takes place at most car dealerships: The customer looks around the showroom; a salesperson approaches the customer and tells him why the car he is interested in is ok, but why another model is even better; the customer takes the car(s) for a test drive; and then the customer, salesperson, and an unseen "manager" engage in a long and drawn out negotiation that, in stories with happy endings, results in the customer getting a reasonable deal on their car of choice.

A Schema Is Like an Organized Tool Rack

Scripts work well for organizing sequences of events. There is another mental representation, called a schema, that does essentially the same thing for information in general. In fact, you could say that a script is a type of schema that is specialized for one particular type of information (i.e., events). 

Traditionally, a schema is described as a structure with labeled slots. I think of it as a tool rack where all the tools have an outline drawn around where they should go. When an old tool is ready to be put away, you find the spot where it fits on the rack and store it there for future use. If an unfamiliar tool comes your way, you will likely look at your well-organized tool rack, see whether the new tool seems like any of the others you already have, and store it with those that are most similar to the new one. 

A good example of a mental schema comes from my hobby-level interest in cars. I love reading about the newest models and their embedded technologies. Over the years, I have built up a schema for cars. The slots in my schema include: category (e.g., sedan, coupe, sports car), make (e.g., Honda, Ford, Porsche), model (e.g., Civic, Taurus, 911), and origin (e.g., foreign or domestic).

Schemas can be violated when you encounter new information that doesn't fit into the existing organizational structure of the schema. When this happens, there are a couple of potential outcomes. One outcome is that you dismiss the new information because it does not fit into your current understanding of the world. Another outcome is that you misconstrue the true nature of the new information by trying to fit it into an existing schema where it doesn't really fit. Yet another outcome of a schema violation is that it inspires a person to modify his or her existing schema to accommodate the new information. I recently had my car schema violated when I learned about the Tesla Type S, which is a purely electric car. It has no tailpipe, spark plugs, or fuel door. In fact, it doesn't even require an oil change! To accommodate these new facts, I had to add a new slot to my car schema, which is fuel type (e.g., gas, pure electric, hybrid).

The STEM Connection

One could argue that the primary goal of education is to help students create accurate and detailed scripts and schemas. Because we live in a changing world, however, educators also need to confer upon their students the tools to notice violations and modify their representations when needed. 

The scientific method is perhaps one of the most successful scripts for producing new knowledge. A scientist starts with noticing something in the world that is in desperate need of an explanation. Toward that end, the scientist formulates a hypothesis and begins the arduous task of collecting data to confirm or disconfirm her hypothesis. This script works well, but it's important to teach budding scientists that they need to be willing (eager even!) to update their beliefs once the data are collected. This can be difficult, of course, but it is the best way to move the field forward. In a sense, scientists need to be flexible in constructing and updating their schemas. 

Some scientific discoveries, like penicillin, happen precisely because some piece of empirical evidence violated a scientist's schema. As the story goes, Alexander Fleming was working on a project that required him to create colonies of the Staphylococcus bacteria. Getting nowhere, he decided to take a vacation, and he left his lab in disarray. As expected, when he got back from vacation, some of his petri dishes were overrun with bacteria. What he didn't expect, however, was that some colonies were suspiciously missing. Fleming noticed that some of the bacteria were killed by a mold that was also growing in the dish. In that moment, Fleming experienced a profound schema violation.

Louis Pasteur famously said, "In the fields of observation, chance favors only the prepared mind." Given the language introduced here, we might recast Pasteur's recommendation to students to become more like Fleming: Take the time to construct a detailed schema. Then, be on the lookout for violations of your schema because they might result in important scientific advancements. 


Share and Enjoy! 

Dr. Bob

For More Information

[1] The answer is: 8, 5, 7, 4, 2, 6, 3, 1, 9

[2] Hollywood relies on schema violations to delight their audiences. The Matrix and Memento are excellent examples of movies that rely on violating our schemas to enhance the storytelling.

A Distinction With a Difference: Declarative and Procedural Knowledge

Representations in Memory: Facts vs. Skills

Let's start with a pop quiz. While answering each question, introspect on what your brain is trying to do. Think about how you are attempting to generate an answer for each question:

1.  What is the capital of Montana?

2.  Solve the following equation in terms of x: 2x + 40 = 80

Do you have some answers? Excellent! Which question was more difficult? Which one took longer to answer? Which question was more factual, and which question was more skill-based? (BTW, if we stuck your head in an MRI and started scanning, we would notice that different brain areas would have become active for the two different questions.) So, what's going on? Why are there differences?

As one of my professors was fond of saying, "If you can imagine three ways the brain can do something, it does it in all ten." The same is true with mental representations. There are several different ways the brain can represent information. For example, we previously talked about the power of mental models and what they afford us. 

While extremely useful – as we saw – mental models are only one way the mind represents knowledge. The mind also stores information using other types of representations, including declarative and procedural memories. A declarative memory is something that can be stated explicitly, like a fact. "Abraham Lincoln was the 16th president of the United States of America" is an example of a declarative fact. Alternatively, a procedural memory is one that encodes how you perform an activity or skill. Knowing how to tie your shoes is a procedural memory that has a muscular component (i.e., a "motor memory"), and knowing how to do long division is an example of a cognitive, procedural skill. What's the point of having different types of memory? What are some of the properties of these different forms of memory [1]?

Properties of Declarative and Procedural Knowledge

The first property has to do with the speed by which these memories are accessed. Declarative memories are slow to access, which makes intuitive sense. Think about how difficult it is sometimes to remember a fact, especially if it's been a really long time since you've thought about that particular subject. Procedural memories, on the other hand, are very fast. You can fire off a procedural skill in very little time, sometimes almost automatically. This automaticity is often a good thing in that it lets you execute complicated skills without having to ruminate over every step required along the way. 

There is a trade off, however. The availability of each representation is differentially influenced by context. Declarative memories are generally available, regardless of the current circumstances. That is, they are independent of the context. You can recall the 16th president anywhere on Earth. Procedural memories, on the other hand, are more sensitive to the current context. For example, it might not occur to you that a specific skill is required when there is nothing in the current environment to cue that skill. Thus, we might describe procedural skills as context dependent (see Table 1).

Table 1. Properties of Procedural and Declarative Memory

Finally, I think it is important to make these distinctions because the representation knowledge changes over time. When you first start to learning something new, the knowledge is generally represented declaratively, but can become procedural over time. Take learning how to drive a car as an example of the process of declarative knowledge becoming proceduralized: 

A lot of what your driving instructor told you was verbal. You had to learn where the gas and brake pedals were. You also had to learn where the turn signal was, the headlights, and all of the various buttons and levers that are required to drive an automobile. You also had to learn a ton of traffic laws, all of which, I'm guessing, were stored as declarative chunks of information. But then, as you became an expert driver (i.e., over the next 10 years), you didn't represent any of that knowledge as explicit, declarative facts. Instead, you no longer had to think about where to place your feet, or what do when changing lanes (i.e., check your mirrors, check your blind spot, signal, etc.). It all became procedural knowledge and happened automatically. The same is true for cognitive tasks. Learning to solve routine problems can eventually become automatized, which means that the declarative representations you had in memory are now so automatic that they are converted into procedural memories [2].

A STEM Example

How does knowing the distinction between declarative and procedural knowledge help us improve education? First, it is has implications for how we train our students. When teaching a new skill, a good approach is to provide a conceptual introduction by articulating a set of declarative chunks of information. As discussed previously, we can increase the odds of storing declarative memories in long-term memory by finding a hook. Connecting the current set of facts to some other piece of knowledge that our students already have gives them a better shot at remembering the new facts later. (I tried to do something similar at the beginning of this post: I assumed that you already knew what a fact and skill were, so I tried to map the concept of declarative [fact] and procedural [skill] knowledge onto those concepts.) 

Once you've motivated the lesson with a conceptual introduction and covered the declarative facts that are needed to develop a new skill, it is time to start deliberately acting on those facts until they are transformed into a skill. Here, it is important to give students plenty of opportunities to put their new-found declarative knowledge into action, accompanied by lots of feedback along the way. That feedback can come from you (as the teacher), other students, tutoring software, or pretty much any source that lets students know when they are on the right track or need a course correction. Again, the analogy to driving a car is pretty good. You need lots of hours behind the wheel, with lots of feedback (from an instructor, mom or dad, a kindly police officer, a not-so-kind driver in the other lane, the curb, the grinding of the gears, etc.) before the declarative facts of how to drive become procedural skill.  

Share and Enjoy! 

Dr. Bob

For More Information

[1] Nokes, T. J., & Ohlsson, S. (2005). Comparing multiple paths to mastery: What is learned? Cognitive Science, 29(5), 769–796.

[2] My favorite example of the automaticity of knowledge is my copy code. When I worked at LRDC, they had a couple of copy machines on each floor. Each person was given a "copy code" which charges the copies that you make to your account. While conducting a study, my research assistant asked me for my code, and I could not verbalize it! I had to "let my figures do the talking." I had to type the code and look at the numbers that I was hitting. The memory was completely converted from a declarative memory to a procedural one. 

Have You Gone Mental?: Mental Models

Using Models to Reason and Infer New Knowledge

Let me ask you a question: How many windows are in your house or apartment? It's entirely possible that nobody's ever asked you this before. At least, that's what I'm banking on. If you've never been asked "How many windows are in your home?", then that means you aren't answering from memory. Instead, the question requires that you compute a value on the spot. How did you accomplish this task?

My prediction is that you visualized your home, and then started a walk-through, counting each window as you moved from room to room. In other words, you used a mental simulation, or a mental model, to answer my question. As it turns out, mental models are great for more than just answering random questions. They are just one instance of a class of mental representations that we use everyday. Mental models are simulations or images that we use to reason about the world and/or infer new knowledge.

What do toilets and light beams have in common? 

Consider another example of a mental model: the flushable toilet. If you know how a toilet works, then you can use your mental model to debug it when things go wrong. For example, a well-constructed mental model will help you figure out why the water keeps running (i.e., the filler float is stuck or the flush valve is stuck in the open position). Or why nothing happens when you depress the handle (i.e., the chain that connects the handle to the flush value fell off or is broken).

In addition to reasoning about the world, mental models are also useful in generating new knowledge through the process of inference. In a previous post, we talked about the power of inheritance to derive new information. This is similar in the sense that you infer new facts by "running" a mental model. 

One of the more famous examples of this is Einstein's claim that he used a mental simulation of riding a beam of light and asking all sorts of questions about what he might observe at that speed. Good thing he interrogated his mental models because it gave birth to the special theory of relativity!

A STEM Example

There are so many examples of mental models in science and engineering that I won't even attempt to catalog them here. In fact, one could argue that STEM education is primarily focused on helping students build detailed and accurate mental models. Here are a couple of illustrative examples.

First, the astute reader probably noticed that a recent post, entitled "Midnight in the Garden of Encoding and Retrieval," attempted to create a mental model of memory. That model proved to be useful when we started asking questions about what happens during encoding, storage, and retrieval. The answers to those questions helped us debug potential reasons why a student might fail to learn a new fact or skill. 

Another example, that I've used in my own research, is the human circulatory system. In one of our studies, we asked about the thickness of the muscle for the right ventricle versus the left ventricle of the heart. If you know that the right side of the heart sends the blood to the lungs, and you know that the lungs are proximal to the heart, then you know it doesn't need to pump very hard; therefore, the muscle in the walls of the right ventricle do not need to be as thick as the muscles in the left ventricle. This is useful knowledge that doesn't need to be taught directly. Instead, it can be inferred by the student through a series of leading questions. 

The final example I will give is one of my favorites [1]. It has to do with the development of the mental model for the Earth. When kids are little, they know, via observation, that the Earth is flat. Later on, they learn that the Earth is round. To make the observation compatible with the authoritative knowledge that they hear from adults, children then reason that the Earth must be round, like a pancake. If you ask them leading questions, such as, "What will happen if you walk for days and days?", they will answer that you will come to the edge of the Earth. 

If the goal of education is to help students develop accurate and complete mental models, then there is a pretty interesting implication for assessment. It is difficult and time-consuming, but developing generative questions is a excellent way to evaluate your students' mental models. Generative questions ask the student to reason about his or her model model. The "muscle thickness of the ventricles" and "walking the Earth for days and days" are good example of generative questions. 

Share and Enjoy! 

Dr. Bob

For More Information

[1] Vosniadou, S., & Brewer, W. F. (1992). Mental models of the Earth: A study of conceptual change in childhood. Cognitive Psychology, 24, 535–585.